Hetero for Stock Prices

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A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of ReturnAuthor(s): Tim BollerslevSource: The Review of Economics and Statistics, Vol. 69, No. 3 (Aug., 1987), pp. 542-547Published by: The MIT PressStable URL: http://www.jstor.org/stable/1925546

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542 THE REVIEW OF ECONOMICS AND STATISTICSA CONDITIONALLY HETEROSKEDASTIC TIME SERIES MODEL FORSPECULATIVEPRICES AND RATES OF RETURN

Tim Bollerslev*Abstract-The distributionof speculativepricechangesandrates of return data tend to be uncorrelatedover time butcharacterizedby volatileand tranquilperiods.A simpletimeseries modeldesignedto capturethisdependenceis presented.The model is an extensionof the AutoregressiveConditionalHeteroskedastic(ARCH)and GeneralizedARCH (GARCH)models obtained by allowingfor conditionallyt-distributederrors.The model can be derived as a simplesubordinatestochasticprocessby includingan additiveunobservableerrorterm in the conditionalvarianceequation.The descriptivevalidityof the modelis illustratedfor a setof foreignexchangeratesandstock priceindices.

L. IntroductionThe distributional properties of speculative prices,stock returns and foreign exchange rates have important

implications for several financial models, includingmodels for capital asset prices and models for thepricing of contingent claims. For example, in empiricaltests of mean-variance portfolio theories, such as theSharpe-Lintner Capital Asset Pricing Model, the vari-ances and covariances of the asset returns are used asmeasures of "dispersion" or "risk." However, depend-ing on the distribution of the returns the variance maynot be a valid or sufficient statistic to use. The distribu-tion of the returns also plays an important role in theBlack-Scholes option pricing formula, and in the pricingof forward contracts in the foreign exchange market.Furthermore, most tests of the efficient markethypothe-sis rely critically on the underlying distributional as-sumptions.Following the seminal works by Mandelbrot (1963)and Fama (1965), where it is shown that the firstdifferences of the logarithm of cotton and commonstock prices generally have fatter tails than are compati-ble with the normal distribution, a voluminous litera-ture has been addressed to this question. Without at-tempting an exhaustive list of these studies, particularlyinteresting are the original papers by Mandelbrot (1963)and Fama (1965) where the stable Paretian family ofdistributions is suggested to characterizethe stochasticproperties of speculative prices. In Praetz (1972) andBlattberg and Gonedes (1974) it is argued that for both

stock priceindices and individualstockpricesthescaledt-distribution,derivableas a continuousvariancemix-ture of normals,has greaterdescriptivevalidity.Othervariancemixturemodelsproposedin the literaturein-clude the compoundeventsmodel by Press(1967), thelognormal-normalmodel by Clark (1973), the sub-ordinatenormal mixturemodel in Westerfield(1977)and the discretemixtureof normaldistributionsin Kon(1984). For a recent list of referenceson the subjectsee also Fielitz and Rozelle (1983) and Boothe andGlassman (1985).The generalconclusionto emergefrom mostof thesestudies is that speculativeprice changesand ratesofreturn are approximatelyuncorrelatedover time andwell describedby a unimodalsymmetricdistributionwith fattertails than thenormal.However,eventhoughthe time series are seriallyuncorrelated,they are notindependent.As notedby Mandelbrot(1963, p. 418),

"..., large changes tend to be followed by largechanges-of eithersign-and smallchangestendtobe followedby smallchanges,.The same regularityhas also been observedfor bothstock pricechangesand foreignexchangeratechanges,see for instance Fama(1965,1970)and Mussa(1979).Indeed,this behaviormightverywellexplainthe recentrejectionof an independentincrementsprocessfor dailystockreturnsin Hinich and Patterson(1985).The Autoregressive Conditional Heteroskedastic(ARCH) model introducedin Engle (1982) explicitlyrecognizesthis typeof temporaldependence.Accordingto theARCH model the conditionalerrordistributionisnormal,but with conditionalvarianceequalto a linearfunction of past squarederrors.Thus, there is a ten-dency for extreme valuesto be followedby other ex-tremevalues, but of unpredictablesign. Furthermore,the unconditionalerrordistributionof theARCHmodelis leptokurtic,reconcilingthe previousempiricalfind-ings.In thepresentpapera simpleextensionof the ARCHmodel to allow for conditionallyt-distributederrorsisgiven.'This developmentpermitsa distinctionbetweenconditionalheteroskedasticityand a conditionallepto-kurtic distribution,eitherof which could accountforthe observed unconditionalkurtosisin the data. In

ReceivedforpublicationDecember9, 1985.Revisionacceptedfor publicationJanuary23, 1987.*NorthwesternUniversity.I am gratefulto Dick Baillie,Rob Engle,Ken Kroner,JeffWooldridgeand two anonymousrefereesfor many helpfulcommentsand suggestionsin improvingan earlierversionofthis paper.Needless to say, I am solely responsiblefor anyremainingerrors.IEngle (1982)recognizesthat otherconditionalerrordistri-butions could be imposed,but to the author'sknowledgenobodyhas yet done thatin practice.

Copyright K)1987


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NOTES 543addition to this extension, we also allow the currentconditional variance to be a function of past conditionalvariances as in the GeneralizedARCH (GARCH) modeldeveloped in Bollerslev (1986).

II. The Standardizedt-distributionSince the econometric model characterizing the dis-tributional properties outlined above applies in a muchbroader context, the discussion in this section will besomewhat general. Thus, let the conditional distributionof yt, t = 1 ... ., T, be standardized t with mean yt-l,variance h,t,l and degrees of freedom v, i.e.,

y= E( ytlA.- 1) + Et = Ytlt-1 + Et

(v + 1 )(v)1( 1/2

X(1 + E2,h1U1(v - 2) 1)) ,v>2 (1)

where 4t-l denotes the a-field generated by all theavailable information up through time t - 1 andft,(Etj4'-t) the conditional density function for Et.The t-distribution is symmetric around 0, and fromKendall and Stuart (1969) the variance and the fourthmoment are equal to

Var(E,t',_j) = htlt-lE( tE41,t_) = 3(v - 2)(v - 4) 1h,1, v > 4.

It is also well known that for 1/v -* 0 the t-distributionapproaches a normal distribution with variance h,l,_l,but for 1/v > 0 the t-distributionhas "fatter tails" thanthe corresponding normal distribution.A particularly appealing featureof the model given in(1) is the fact that it can be derived as a subordinatestochastic process from the conditional normal model:

yt = E(y,t',i_) + E, = Ytlt-l + Et,1t't-,1, Ut - (2qh,)Th) 2exp(_1/2tht 1), (2)

where ut denotes the prediction errorin the conditionalvariance equationh, = E(h,tli,1) + ut = h,t,ll + ut. (3)

The decomposition in (3) is analogous to the decom-position in (1) of yt into its conditional expectation,yt,t,_ and a prediction error, E,. Note, however, thatunlike yt, the conditional variance at time t, h,, is ingeneral unobservable. Specifying the conditional distri-bution of ut to be a transformed inverted gamma-i

distribution:u,A, l 1 vg/2 ( -t l)2= -2 1)htlt1) (ut + h,l,_1)

xexp((1 - j)htlt,1(Ut + hti,-1 ) (4)ut > -h,l,_l, v > 2,

and following the same line of arguments as in Raiffaand Schlaifer (1961), it is possible to show that thegJ(u,tli,1) mixture of cA4',,-,ut equals the standard-ized t-distribution given in (1).

Let 0 denote all the unknown parameters in model(1), including the degree of freedom parameter P.By theprediction error decomposition the loglikelihood func-tion for the sample YI ... IYT is then, apart from someinitial conditions, given byT

LT(O) = E log f(,It',_1),t=1and standard inference procedures regarding 0 are im-mediately available.However, when testing against the interesting nullhypothesis of conditionally normal errors, i.e., 1/v = 0,1/v is on the boundary of the admissible parameterspace, and the usual test statistics will likely be moreconcentrated towards the origin than a X1 distribution.Some preliminary Monte Carlo evidence confirms thisconjecture. For moderately sized samples the correct 5%critical value for the Likelihood Ratio (LR) test statisticfor the null hypothesis 1/v = 0 is approximately 2.7.Also, the bias in the maximum likelihood estimates forl/v is very small for sample sizes one hundred orlarger. For a more detailed discussion of the MonteCarlo findings the readeris referredto Bollerslev (1985)and Blattberg and Gonedes (1974) for a similar smallscale simulation study.

III. The GARCH(p,q) ModelAs discussed above, several studies indicate that the

change in speculative prices and rates of return areapproximately uncorrelatedover time, but characterizedby tranquil and volatile periods. To allow for such adependence we shall here take the conditional mean,Ytlt-1, as constant,

Yt = / + Et (5)along with a GARCH(p, q) model for the conditionalvariance

E( E,2lpt_ 1) =htlt-l1q P

= w ? E ai1tc-i ? E Ay^.-JIt-z=1 1=1 (6)


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544 THE REVIEW OF ECONOMICS AND STATISTICSwhere o > 0, ax > 0, ij2 0. It is obvious that in themodel given by (5) and (6) there is a tendency for large(small) residuals to be followed by other large (small)residuals but of unpredictable sign.

Rearranging terms in (6), the GARCH(p, q) model isreadily interpreted as a generalization of Engle's (1982)ARCH(q) model obtained by including p "movingaverage terms" of the form (E2 - hi in theautoregressive equation for the conditional expectationof c2, cf. Bollerslev (1986). The orders of p and q cantherefore be identified by applying the traditional Boxand Jenkins (1976) time series techniques to the auto-correlations and partial autocorrelations for the squaredprocess, c2 (see Bollerslev (1987) for a more detaileddiscussion along these lines).Even though the unconditional distribution corre-sponding to the GARCH(p, q) model with condition-ally normal errors is leptokurtic, it is not clear whetherthe model sufficiently accounts for the observed lepto-kurtosis in financial time series. In particular, a "fat-tailed" conditional distribution might be superior to theconditional normal. The econometric model presentedin the previous section with conditionally t-distributederrors, (1), combined with (5) and (6) allows for thispossibility.

IV. EmpiricalExample and Concluding RemarksThe first set of data consists of daily spot prices fromthe New York foreign exchange market on the U.S.

dollar versus the British pound and the Deutschmarkfrom March 1, 1980 until January 28, 1985 for a total of1,245 observations excluding weekends and holidays.2The spot prices, St, are converted to continuously com-pounded rates of return,yt = log,(S/St-1)

As documented by many previous studies the yt seriesare approximately uncorrelatedover time.In particular, for the British pound the Ljung-Box(1978) portmanteau test statistic for up to tenth orderserial correlation in (yt - ,u) takes the value Q(10) =7.64, cf. table 1, which is not significant at any reason-able level in the corresponding asymptotic X2o distribu-tion. On the other hand, (yt - ,U)2 is clearly not uncor-related over time, as reflected by the highly significantLjung-Box test statistic for absence of serial correlationin the squares, Q2(10) = 74.79 distributed asymptoti-cally as a X2 distribution, see McLeod and Li (1983).This absence of serial dependence in the conditionalfirst moments along with the dependence in the condi-

tional second moments is one of the implications of theGARCH(p, q) model given by (5) and (6). Further-more, the unconditional sample kurtosis for the Britishpound K = 4.81, cf. table 1, exceeds the normal value ofthree by several asymptotic standard errors, A24/1244= .139. Of course, this is also in accordance with theimplications of the GARCH(p, q) model.As mentioned above, the appropriate orders for pand q could now be identified by standard Box-Jenkinsmethodology applied to the squared residuals, (yt - ,I)2,cf. Bollerslev (1987) and Engle and Bollerslev (1986).However, for illustrative purposes we shall here con-sider the simple GARCH(1, 1)-t model only:Yt = IL + Et

htl_ = c + a 21 + I3htlIt2Et.l.- 1 - f,"( 'E.1+.- 1) (7)

As we shall see, it just so happens that this particularlysimple model fits the time series chosen here quiteadequately.Maximum likelihood estimates of the parameters inmodel (7) are presented in table 2 along with asymptoticstandard errors in parentheses. The estimates are ob-tained by the Berndt, Hall, Hall and Hausman (1974)algorithm using numerical derivatives.The Ljung-Box test statistic for the standardized re-siduals, 1,ht-l,t-I and the standardized squared residu-als, E,2hI,,- from the estimated GARCH(1, 1)-t modeltake the values Q(10) = 4.34 and Q2(10) = 8.60, re-spectively, cf. table 1, and thus do not indicate anyfurther first or second order serial dependence. It is alsointeresting to note that the implied estimate of theconditional kurtosis, 3(v - 2)(v - 4)' = 4.45, cf. sec-tion II, is in close accordance with the sample analoguefor E,ht,lt1, K = 4.63.

TABLE 1.- SUMMARY STATISTICSY,-1t GARCH(1,1)-t

Q(10) Q2(10) K Q(10) Q2(10) KExchange RatesBritain 7.64 74.79 4.81 4.34 8.60 4.63Germany 9.16 133.08 4.21 7.70 9.51 3.76Stock Price Indices500 Composite 15.00 26.36 4.06 14.08 8.08 3.89Industrial 14.87 25.03 3.98 14.11 7.75 3.84

Capital Goods 9.28 17.92 4.17 8.84 9.29 4.06Consumer Goods 19.20 49.03 5.70 12.81 9.27 4.32Public Utilities 26.57 115.51 5.41 19.61 7.69 4.08Note: Q(10) and Q2(10) denote the Ljung-Box (1978) portmanteau testsfor up to tenth order serial correlation in the levels and the squares, respec-tively. K is the usual measure of kurtosis given by the fourth sample momentdivided by the square of the second moment.

2The data were kindly provided by Dick Baillie.


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NOTES 545TABLE2.-GARCH(1, 1)-t MAXIMUMLIKELIHOODESTIMATESAND TESTS

1 a Ai 1/v LR11P=o LRO=O=?Exchange RatesBritain -4.56 10-4 0.96 10-6 .057 .921 .123 41.26 65.21(1.67 10-4) (0.46 .10-6) (.017) (.023) (.024)Germany -5.86 10-4 0.13 10-6 .095 .881 .072 13.92 101.77(1.78 10-4) (0.59 * 10-6) (.021) (.027) (.022)Stock Price Indices500 Composite 5.63 10-3 0.17 10-3 .074 .768 .139 12.11 7.03(1.46 10-3) (0.13 10-3) (.045) (.148) (.055)Industrial 5.73 10-3 0.19 10-3 .078 .756 .129 10.46 6.87(1.51 10-3) (0.14 10- 3) (.046) (.150) (.054)Capital Goods 5.78 10-3 0.27 10-3 .065 .751 .140 13.66 4.08(1.73 10-3) (0.23 10-3) (.048) (.183) (.052)Consumer Goods 5.53 10-3 0.14 10-3 .103 .782 .161 19.01 19.75(1.48 10-3) (0.07 10-3) (.041) (.081) (.052)Public Utilities 4.32 10-3 0.05 10-3 .123 .820 .131 11.78 37.82(1.27 10-3) (0.03 10-3) (.047) (.066) (.049)

Note: Asymptotic standard errors are in parentheses.

This estimate of the conditional kurtosis differs sig-nificantly from the normal value of three, as seen by theLRI,,==0 test for the GARCH(l, 1) model with condi-tionally normal errors, equal to 41.26. Note, the MonteCarlo results mentioned in section II indicate that inthis situation the LR test statistic is more concentratedtowards the origin than a x2 distribution, and thereforeevaluating LRI,7=0 in the x2 distribution leads to aconservative test for conditional normality, implyingrejection at even higher levels.On the other hand, the standardized t-distributionwith constant conditional variance which has previouslybeen suggested in the literature to characterize the dis-tributional properties of foreign exchange rates, cf.Rogalski and Vinso (1978), fails to take account of theconditional dependence in the second moments. Indeed,the LR a=0= test statistic equals 65.21, cf. table 2,which is highly significant at any level in the corre-sponding asymptotic X2 distribution(see Engle, Hendryand Trumble (1985) for a discussion of the small sampleproperties of ARCH tests and estimators).The results for the Deutschmark are quite similar tothe results for the British pound discussed above. Again,the GARCH(1, 1)-t seems to provide a simple andparsimonious description of the time series properties.The Ljung-Box test statistics do not indicate any furtherfirst or second order dependence in the standardizedresiduals. The estimated value of the conditional kurto-sis, 3(vi - 2)(v - 4)-1 = 3.61, is also in accordancewiththe sample analogue of %j,4h,-I1,K = 3.76. The LRa=otest for absence of conditional heteroskedasticityis equalto 101.77 and highly significant at any level, as is theLRI ,=o0test for GARCH(1, 1) with conditionally nor-mal errors equal to 13.92.

Note, the estimated values for a + ,B are close to onefor both currencies. Thus, an Integrated GARCH(1, 1)- t model imposing the restriction a + ,B= 1 mightprovide an even simpler characterizationof the foreignexchange rates considered here. However, when a + ,8= 1 the unconditional variance does not exist, and theasymptotic properties of the maximum likelihood esti-mators are not clear. See Engle and Bollerslev (1986) fora precise definition of the IGARCH model, along with adiscussion of the statistical problems that arise in thatsituation.It would of course be interesting to see whether theresults presented above carry over to other currenciesand sampling intervals. The recent empirical findings inMilh0j (1987), Hsieh (1985) and McCurdy and Morgan(1985) are all suggestive. Even though the unconditionalerror distribution corresponding to the ARCH andGARCH models with conditionally normal errors areleptokurtic, they find that the models do not fullyaccount for the leptokurtosis in the different foreignexchange rates analyzed.We now turn to the second set of data which includesfive different monthly stock price indices for the U.S.economy, namely Standard and Poor's 500 Composite,Industrial, Capital Goods, Consumer Goods and PublicUtilities price indices. The indices are monthly averagesof daily prices from 1947.1 to 1984.9, i.e., 453 observa-tions.3 It is well known that if the daily price changesare uncorrelated, averaging daily prices to obtain anaverage monthly price index, P, induces a spurious firstorder moving average correlation structurewith movingaverage parameter close to 0.268 corresponding to a

The data were taken from the CitibankEconomicDatabase.


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546 THE REVIEW OF ECONOMICS AND STATISTICSfirst order correlation coefficient equal to 0.250; cf.Working (1960). Thus, following Praetz (1982) we definethe rate of return series for the stock price indices, Yt,tobe

y= (1 + .268B)1loge(P,/P,_1), (8)where B denotes the usual backshift operator.4From rows 3 through 7 in table 1 it is clear that eventhough the rates of returnseries tend to be uncorrelatedover time, there is again a tendency for large and smallresiduals to cluster togetheras seen by the highly signifi-cant Q2(10) statistics in column 2. The estimates of theGARCH(1, 1)-t model reported in the last five rows intable 2 also reflect this fact. Most of the coefficients aresignificant at traditional levels. Furthermore, none ofthe Ljung-Box portmanteau tests for the standardizedresiduals, ',h7,it/I reported in table 1 are significant atthe usual 5% level, except for Q(10) for Public Utilities.It is interesting to note that all of the LR,,10=o tests forthe standardized t-distribution, which have previouslybeen suggested in the literature to characterize the sto-chastic behavior of stock price indices, cf. Praetz (1972),are significant at the 5% level or lower, except forCapital Goods. The LR1/^=0 tests for the GARCH(1, 1)model with conditionally normal errors, are significantat the 1% level or lower using the conservation x2distribution.

Summing up, the results presented in this sectionconfirm the previous findings in the literature thatspeculative price changes and rates of return series areapproximately uncorrelated over time but characterizedby tranquil and volatile periods. The standardized t-dis-tribution fails to take account of this temporal depen-dence, and the ARCH or GARCH models with condi-tionally normal errors do not seem to fully capturethe leptokurtosis.. Instead, the relatively simpleGARCH(1, 1)-t model fits the data series consideredhere quite well. Of course, it remains an open questionwhether other conditional error distributions provide aneven better description. Another interesting question iswhether high order GARCH models might be called forwhen modelling other financial time series. The em-pirical relevance of the IGARCH model also deservesfurther investigation. We leave the answer to all of thesequestions for future research.

4In practice, the expansion of (1 + .268B)-l was truncatedafter its first four terms, 1 - .2680B + .0718B2 -.0192B3 +.0052 B4.

REFERENCESBerndt, Ernst K., Bronwyn H. Hall, Robert E. Hall, and JerryA. Hausman, "Estimation and Inference in NonlinearStructural Models," Annals of Economic and SocialMeasurement 4 (1974), 653-665.

Blattberg, Robert C., and Nicholas J. Gonedes, "A Compari-son of the Stable and Student Distributions as Statisti-cal Models for Stock Prices," Journal of Business 47(Apr. 1974), 244-280.Bollerslev, Tim, "A Conditionally Heteroskedastic Time SeriesModel for Security Prices and Rates of Return Data,"UCSD Discussion Paper 85-32 (1985).

______ "Generalized Autoregressive Conditional Hetero-skedasticity," Journal of Econometrics 31 (June 1986),307-327._____ "On the Correlation Structure for the GeneralizedAutoregressive Conditional Heteroskedastic Process,"Journal of Time-Series Analysis (forthcoming 1987).Boothe, Paul, and Debra Glassman, "The Statistical Distribu-tion of Exchange Rates: Empirical Evidence and Eco-nomic Implications," University of Alberta, Depart-ment of Economics, Research Paper No. 85-22 (1985).Box, George E. P., and Gwilym M. Jenkins, Time SeriesAnalysis: Forecasting and Control, revised edition (SanFrancisco: Holden Day, 1976).Clark, Peter, "A Subordinate Stochastic Process Model with

Finite Variance for Speculative Prices," Econometrica41 (1) (1973), 135-155.Engle, Robert F., "Autoregressive Conditional Heteroscedas-ticity with Estimates of the Variance of U.K. Inflation,"Econometrica 50 (1982), 987-1008.Engle, Robert F., and Tim Bollerslev, "Modelling the Per-sistence of Conditional Variances," Econometric Re-views 5 (1) (1986), 1-50.Engle, Robert F., David F. Hendry, and David Trumble,"Small-Sample Properties of ARCH Estimators andTests," Canadian Journal of Economics 18 (1) (1985),66-93.Fama, Eugene F., "The Behavior of Stock Market Prices,"Journal of Business 38 (Jan. 1965), 34-105.____, "Efficient Capital Markets, A Review of Theory andEmpirical Work," Journal of Finance 25 (May 1970),383-417.Fielitz, Bruce D., and James P. Rozelle, "Stable Distributionsand the Mixture of Distributions Hypothesis for Com-mon Stock Returns," Journal of the American StatisticalAssociation 78 (Mar. 1983), 28-36.Hinich, Melvin J., and Douglas M. Patterson, "Evidence ofNonlinearity in Daily Stock Returns," Journal of Busi-ness and Economic Statistics 3 (1) (1985), 69-77.Hsieh, David A., "The Statistical Properties of Daily ForeignExchange Rates: 1974-1983," University of Chicago,Graduate School of Business, manuscript (1985).Kendall, Maurice G., and Alan Stuart, The AdvancedTheoryofStatistics, Vol. 1, 3rd ed. (New York: Hafner PublishingCompany, 1969).Kon, Stanley J., "Models of Stock Returns-A Comparison,"Journal of Finance 39 (1) (1984), 147-165.Ljung, Greta M., and George E. P. Box, "On a Measure ofLack of Fit in Time Series Models," Biometrika 65(1978), 297-303.Mandelbrot, Benoit, "The Variation of Certain SpeculativePrices," Journal of Business 36 (Oct. 1963), 394-419.McCurdy, Thomas H., and Ieuan G. Morgan, "Testing theMartingale Hypothesis in the Deutschmark/US DollarFutures and Spot Markets," Queen's University, De-partment of Economics, Discussion Paper No. 639(1985).McLeod, A. J., and W. K. Li, "Diagnostic Checking ARMATime Series Models Using Squared-ResidualAutocorre-lations," Journal of Time Series Analysis 4 (4) (1983),269-273.


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NOTES 547Milh0j, Anders, "A Conditional Variance Model for DailyDeviations of an Exchange Rate," Journal of Businessand Economic Statistics 5 (1) (1987), 99-103.Mussa, Michael, "Empirical Regularities in the Behavior ofExchange Rates and Theories of the Foreign ExchangeMarket," Carnegie-RochesterConferenceSeries on Pub-lic Policy 11 (1979), 9-57.Praetz, P. D. "The Distribution of Share Price Changes,"Journal of Business 45 (Jan. 1972), 49-55., "The Market Model, CAPM and Efficiency in theFrequency Domain," Journal of Time Series Analysis 3

(1) (1982), 61-79.Press, S. J. "A Compound Events Model of Security Prices,"

Journal of Business 40 (July 1967), 317-335.Raiffa, Howard, and R. Schlaifer, Applied Statistical DecisionTheory (Cambridge: HarvardUniversity Press, 1961).Rogalski, Richard J., and J. D. Vinso, "Empirical Properties ofForeign Exchange Rates," Journal of InternationalBusi-ness Studies 9 (1) (1978), 69-79.Westerfield, R., "The Distribution of Common Stock PriceChanges: An Application of Transactions Time andSubordinate Stochastic Models," Journal of Financialand QuantitativeAnalysis 12 (1977), 743-765.Working, Holbrook, "Note on the Correlation of First Differ-ences of a Random Chain," Econometrica 28 (4) (1960),916-918.

THE LINK BETWEEN THE U.S. DOLLAR REAL EXCHANGE RATE, REALPRIMARY COMMODITY PRICES, AND LDCS' TERMS OF TRADE

Agathe Cote *Abstract-The correlationbetweena real appreciationof theU.S. dollarand a declinein U.S. relativeprimarycommodityprices does not imply that such an appreciationleads to adeteriorationin the termsof tradeof commodityproducingLDCs. In fact, empiricalresultsreportedin this note suggestthat the dollar appreciationhas had a beneficialimpact onthesecountries'termsof trade.

I. IntroductionThe behaviour of non-oil primary commodity pricesover recent years has induced observers to examine the

role of "non-traditional" factors in the determination ofprice movements. In this context, a variable that hasgained emphasis is the U.S. dollar effective exchangerate. Movement in the U.S. dollar exchange ratevis-a-vis other major currencies has an impact on com-modity prices expressed in U.S. dollars since it in-fluences the relative prices of substitutes or productionfactors.In this note we examine the link between the U.S.dollar real effective exchange rate, industrial world realprices of non-oil primary commodities (prices of com-modities relative to prices of manufactured goods ex-ported by industrial countries), and LDCs' terms oftrade. We argue that this link cannot be predicteda priori. Contrary to Dornbusch (1983), our empiricalresults suggest that the real appreciation of the U.S.

dollar vis-a-vis other industrial countries' currencies hasled to an improvement in the LDCs' terms of trade.The remainder of the paper is organized in the follow-ing way. In section II we examine the theory, while insection III we discuss our regression results. Conclu-sions are drawn in the last section.II. Theoretical Considerations

An analysis of the link between the U.S. dollar effec-tive exchange rate and U.S. prices of primary commod-ities has been conducted in nominal terms by Sachs(1985, Appendix), and in real terms by Dornbusch(1985a, b, c). The argument is the following. Assumethat the "law of one price" holds for primary commod-ities and that prices of manufactured goods (used as thedeflator) are fixed in local currency terms both in theUnited States and abroad. Demand and supply of com-modities in each area depend on the local relative priceof commodities. A real appreciation of the U.S. dollarvis-a-vis currencies of major industrial countries im-mediately leads (with the dollar prices of commoditiesunchanged) to an increase in the local relative price ofcommodities in other industrial countries. Foreign de-mand for commodities is therefore reduced (foreignsupply should increase as well), which should lead to adecrease in their equilibriumdollar price. The extent ofthe decline in dollar commodity prices depends on theweight of the United States as a producer and consumerof primary commodities. The more important is theUnited States, the less is the U.S. dollar price adjust-ment.

Therefore, one can expect that a real appreciation ofthe U.S. dollar vis-a-vis currencies of major industrialcountries leads to a decrease in the real price of com-

Received for publication April 15, 1985. Revision acceptedfor publication November 25, 1986.* Bank of Canada.The views expressed in this paper are those of the author andno responsibility for them should be attributed to the Bank ofCanada. The author would like to thank David Longworth forhis very helpful comments. Any remaining errors are theresponsibility of the author.

Copyright ?O1987

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Hetero for Stock Prices